The discrete magnetic Laplacian: geometric and spectral preorders with applications by John Stewart Fabila Carrasco A dissertation submitted by in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Mathematical Engineering UNIVERSIDAD CARLOS III DE MADRID Advisor: Fernando Lledó Madrid, June 2020. 2 This document is under terms of "Creative Commons license Attribution - Non Commercial - Non Derivatives ". 3 Published and submitted content The main results of this thesis are based on the next list of papers. • J. S. Fabila-Carrasco, F. Lledó, and O. Post. Spectral gaps and discrete magnetic Laplacians. Linear Algebra and its Applications 547 (2018): 183-216. https://doi.org/10.1016/j.laa.2018.02.006 I am an author of this work, and it is included in Chapter 2, 3 and 4. The material from this source included in this thesis is not singled out with typographic means and references. • J. S. Fabila-Carrasco and F. Lledó. Covering graphs, magnetic spectral gaps and applications to polymers and nanoribbons. Symmetry 11(9) (2019): 1163. https://doi.org/10.3390/sym11091163 I am an author of this work, and it is included in Chapter 2, 3 and 4. The material from this source included in this thesis is not singled out with typographic means and references. • J. S. Fabila-Carrasco, F. Lledó, and O. Post. Spectral order of discrete weighted graphs. In preparation. • J. S. Fabila-Carrasco, F. Lledó, and O. Post. A construction of isospectral graphs. In preparation. 4 The previous papers were presented by the author of this dissertation in the next following talks: • Q-Math Seminar at Universidad Carlos III de Madrid. Spectral gaps and the magnetic Laplacian on graphs. March 2017. https://aplicaciones.uc3m.es/publicacion/agenda/show?id=30895 • 11th Workshop of Young Researches in Mathematics at Universidad Complutense de Madrid. Spectral gaps and discrete magnetic Laplacians. September 2017. http://www.mat.ucm.es/imi/YoungResearchers17/Abstracts/Abstract_JSFabilaCarrasco.pdf • Q-Math Seminar at Universidad Carlos III de Madrid. Comparing graphs using the spectrum of the magnetic Laplacian. January 2018. http://www.q-math.es/Other.php?year=2017/2018 • Analysis Seminar at the University of Trier. Spectral ordering of discrete weighted graphs. September 2018. • Analysis Seminar at the University of Trier. Construction of families of isospectral graphs. November 2018. • Discrete Mathematics Seminar at Universidad de Valladolid. Construction of cospec- tral graphs. June 2019. http://smd2019.blogs.inf.uva.es/programa/ • 13th Workshop of Young Researches in Mathematics at Universidad Complutense de Madrid. Construction of cospectral graphs. September 2019. http://blogs.mat.ucm.es/mmatavan/wp-content/uploads/sites/12/2019/09/cartel-WorkshopJovenes2019F.pdf Also, a simple code for the software Wolfram Mathematica was written to compute the eigenvalues for any finite magnetic weighted graph. The code is presented in AppendixB and it is available in the git repository: https://github.com/JohnFabila/Magnetic-Laplacian. 5 Agradecimientos Quiero agradecer a mi tutor Fernando Lledó por haber hecho posible esta tesis. Me refiero no solo al trabajo en conjunto para desarrollar las matemáticas presentadas en este trabajo, sino también al apoyo personal a lo largo de estos años, para conmigo y para con mi familia. Gracias. Un agradecimiento muy especial al Dr. Olaf Post por involucrarse tan activamente en este trabajo, por recibirme en la Universidad de Trier y hacer mi estancia junto con la de mi familia lo más placentera posible en esta bella ciudad. Gracias a su esposa Linda y a su hijo Felix por todas esas comidas y paseos. Esos tres meses en Alemania han sido de lo más maravillosos. Danke Olaf. Gracias a la Universidad Carlos III de Madrid y a todo el personal del Departamento de Matemáticas por el apoyo que me brindaron en estos años y a todos mis compañeros de despacho a lo largo de este tiempo. Un agradecimiento a Carlos Marijuán, por su involucramiento y sus comentarios, que permitieron mejorar notablemente la notación del presente trabajo. A mi familia por su apoyo, lo sentía tan cercano a pesar de los 9,000 km. de distancia. Gracias a mis padres: Agustina y Efraín. A mis hermanas: Gwendy y Debbie. A mis sobrinas: Daniela, Andrea y a Dieguito bebé. Finalmente, gracias a Are por ser la mejor compañera de esta vida y a Sofia, por ser el motor de todo. Gracias a ellas por su amor, su comprensión y cariño que me dieron las fuerzas necesarias para llegar hasta aquí. Mencionar a todas las personas que me han apoyado en mi formación profesional y personal para la culminación de esta tesis doctoral es una tarea imposible. Gracias totales... Este trabajo fue apoyado por el Ministerio de Economía y Competitividad mediante los programas DGI MTM2014-54692-P y MTM2017-84098-P. También se recibió la ayuda de 6 movilidad modalidad B en el 2018 para la realización de una estancia de tres meses en la Universidad de Trier, Alemania. 7 Para el mundo de Sofia y para todos esos héroes desconocidos... 9 11 Contents 0 Introduction 13 1 Graph Theory and Discrete Analysis 25 1.1 Graph Theory...................................... 26 1.1.1 Families of Graphs.............................. 28 1.1.2 Basic Operations with Graphs....................... 29 Delete edges.................................. 29 Contracting vertices............................. 30 1.2 Weighted Graphs.................................... 31 1.2.1 Examples of weights.............................. 32 1.3 Magnetic Graphs.................................... 33 1.4 Magnetic Weighted Graphs............................. 35 1.5 Discrete Magnetic Laplacian............................. 38 1.6 Spectral Graph Theory................................ 41 1.6.1 Bipartiteness and the spectrum...................... 42 2 Graphs and preorders 45 2.1 Geometric preorder ⊑ for infinite MW-graph.................. 45 2.2 Spectral preorder ≼ for finite MW-graphs..................... 49 2.3 Geometric preorder ⊑ implies spectral preorder ≼ for finite MW-graphs. 52 2.4 Geometric perturbations and elementary operations............. 55 2.4.1 Deleting an edge............................... 55 2.4.2 Vertex contraction............................... 59 2.4.3 Virtualising edges and vertices...................... 62 3 Periodic and Covering Graphs 67 3.1 Covering graphs.................................... 68 3.2 Periodic graphs..................................... 69 3.3 Discrete Floquet Theory............................... 72 3.4 Vector Potential as a Floquet Parameter...................... 74 12 4 Application I: Spectral gaps 77 4.1 Spectral gaps...................................... 78 4.1.1 Magnetic spectral gaps on finite graphs................. 78 4.1.2 Spectral gaps in G-periodic graphs.................... 84 4.2 Examples......................................... 87 4.2.1 Periodic graph without magnetic field.................. 88 4.2.2 Periodic graph with periodic magnetic potential........... 90 5 Application II: Isospectral graphs 99 5.1 A motivating class of examples........................... 101 5.2 Geometric construction of isospectral graphs and partitions of a natural number.......................................... 104 5.3 Examples of isospectral graphs with magnetic potentials........... 113 6 Miscellaneous Applications 117 6.1 Composite operations................................. 117 6.1.1 Edge contraction............................... 117 6.1.2 Delete a vertex................................. 120 6.2 Spectral graph theory and combinatorics..................... 122 6.2.1 Cheeger constants and frustration index................ 124 6.2.2 Algebraic connectivity and spanning trees............... 127 6.3 Chemical graph theory................................ 128 7 Conclusions and future work 131 A Discrete Floquet Theory 135 A.0.1 The Fourier transformation......................... 136 A.0.2 Discrete Floquet theory on graphs..................... 138 B Program 143 B.0.1 Code....................................... 143 List of Abbreviations 147 List of Symbols 149 List of Figures 151 References 153 Alphabetical Index 161 13 0 Introduction Graph Theory is an important part of discrete mathematics with increasingly strong links to other branches. There are important connections with pure mathematics like Algebra [Big93; GR01], Combinatorics [LL00], Geometry [AL95; PR00], Group The- ory [DD89; Lub94], Riemannian Manifolds [Maj13] and graph-like spaces [Pos12], to mention only a few. For example, a graph can be seen as a discrete structure resulting from a limit of specific continuous structure like, e.g., a Riemannian Manifold [Sin06]. The graph can have additional structure, for example, metric graphs [BC08; LP08a; Pos12] or a metric graph together with a self-adjoint differential operator, known as quantum graph [BK13; Kuc04; Kuc05] (see also references therein). However, its interest does not come only from a theoretical, but also from an applied point of view. Many problems in real-life can be modelled by a diagram (graph) that consists of a set of points (vertices) together with some lines joining pairs of points (edges). Some applications of graph theory are, for example, in social-networks [HRH02], chemistry [Jan+07] or computer science [Deo74]. Graphs are also interesting spaces on which one can do analysis. This branch of the mathematics is known as spectral graph theory [Bap10; Chu97; Exn+08; Mer94]. The analysis on graphs is a relatively new and active area of research. In the last years, the